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Harmonic oscillators. One of the simplest yet most important second-order, linear, constant- coefficient differential equations is the equation for a harmonic oscilator. This equation models the motion of a mass attached to a spring. The spring is attached to a vertical wall and the mass is allowed to slide along a horizontal track. We let z denote the displacement of the mass from its natural resting place (with x > 0 if the spring is stretched and x < 0 if the spring is compressed). Therefore the velocity of the moving mass is a' (t) and the acceleration is a"(t). The spring exerts a restorative force proportional to r(t). In addition, there is a frictional force proportional to r' (t) in the direction opposite to that of the motion. There are three parameters for this system: m denotes the mass of the oscillator, b>0 is the damping constant, and k> 0 is the spring constant. Newton's law states that the force acting on the oscillator is equal to mass times acceleration. Therefore the differential equation for the damped harmonic oscillator is mx" + bx' + kx = 0. (1) Lun |_min Assume the mass m = 1. (a) Transform Equation (1) into a system of first-order equations. (b) For which values of k, b does this system have complex eigenvalues? Repeated eigenvalues? Real and distinct eigenvalues? (c) Find the general solution of this system in each case. (d) Describe the motion of the mass when the mass is released from the initial position x = 1 with zero velocity in each case.

Harmonic oscillators. One of the simplest yet most important second-order, linear, constant- coefficient differential equations is the equation for a harmonic oscilator. This equation models the motion of a mass attached to a spring. The spring is attached to a vertical wall and the mass is allowed to slide along a horizontal track. We let z denote the displacement of the mass from its natural resting place (with x > 0 if the spring is stretched and x < 0 if the spring is compressed). Therefore the velocity of the moving mass is a' (t) and the acceleration is a"(t). The spring exerts a restorative force proportional to r(t). In addition, there is a frictional force proportional to r' (t) in the direction opposite to that of the motion. There are three parameters for this system: m denotes the mass of the oscillator, b>0 is the damping constant, and k> 0 is the spring constant. Newton's law states that the force acting on the oscillator is equal to mass times acceleration. Therefore the differential equation for the damped harmonic oscillator is mx" + bx' + kx = 0. (1) Lun |_min Assume the mass m = 1. (a) Transform Equation (1) into a system of first-order equations. (b) For which values of k, b does this system have complex eigenvalues? Repeated eigenvalues? Real and distinct eigenvalues? (c) Find the general solution of this system in each case. (d) Describe the motion of the mass when the mass is released from the initial position x = 1 with zero velocity in each case.

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
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Harmonic oscillators. One of the simplest yet most important second-order, linear, constant- coefficient differential equations is the equation for a harmonic oscilator. This equation models the motion of a mass attached to a spring. The spring is attached to a vertical wall and the mass is allowed to slide along a horizontal track. We let z denote the displacement of the mass from its natural resting place (with x > 0 if the spring is stretched and x < 0 if the spring is compressed). Therefore the velocity of the moving mass is a' (t) and the acceleration is a"(t). The spring exerts a restorative force proportional to r(t). In addition, there is a frictional force proportional to z' (t) in the direction opposite to that of the motion. There are three parameters for this system: m denotes the mass of the oscillator, b>0 is the damping constant, and k> 0 is the spring constant. Newton's law states that the force acting on the oscillator is equal to mass times acceleration. Therefore the differential equation for the damped harmonic oscillator is mx" + bx' + kr = 0. (1) k Lui Assume the mass m = 1. (a) Transform Equation (1) into a system of first-order equations. (b) For which values of k, b does this system have complex eigenvalues? Repeated eigenvalues? Real and distinct eigenvalues? (c) Find the general solution of this system in each case. (d) Describe the motion of the mass when the mass is released from the initial position z = 1 with zero velocity in each case.
Transcribed Image Text:Harmonic oscillators. One of the simplest yet most important second-order, linear, constant- coefficient differential equations is the equation for a harmonic oscilator. This equation models the motion of a mass attached to a spring. The spring is attached to a vertical wall and the mass is allowed to slide along a horizontal track. We let z denote the displacement of the mass from its natural resting place (with x > 0 if the spring is stretched and x < 0 if the spring is compressed). Therefore the velocity of the moving mass is a' (t) and the acceleration is a"(t). The spring exerts a restorative force proportional to r(t). In addition, there is a frictional force proportional to z' (t) in the direction opposite to that of the motion. There are three parameters for this system: m denotes the mass of the oscillator, b>0 is the damping constant, and k> 0 is the spring constant. Newton's law states that the force acting on the oscillator is equal to mass times acceleration. Therefore the differential equation for the damped harmonic oscillator is mx" + bx' + kr = 0. (1) k Lui Assume the mass m = 1. (a) Transform Equation (1) into a system of first-order equations. (b) For which values of k, b does this system have complex eigenvalues? Repeated eigenvalues? Real and distinct eigenvalues? (c) Find the general solution of this system in each case. (d) Describe the motion of the mass when the mass is released from the initial position z = 1 with zero velocity in each case.
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